A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Convergence of stochastic processes via convergence of infinitesimal generators

Given a sequence of sequence processes $(X_N(\cdot))_{N \geq 0}$, I want to show this sequence converges to another process $X(\cdot)$ by considering that the sequence of generators $(A_N)_{N \geq 0}$ ...
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32 views

A counterexample for supremum of stopping times

Let $\mathbb{F} = \{ \mathcal{F}_t \}_{t \geq 0}$ be a continuous time filteration. $\tau : \Omega \to [0, \infty]$ is called an $\mathbb{F}$-stopping time if $\{ \tau \leq t \} \in \mathcal{F}_t$ for ...
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11 views

Distribution of Hitting Times

I am curious about the following problem: Let the diffusion process $\{X_t\}_{t\ge 0}$ be defined as $$dX_t=c(1-X_t)X_td B_t$$ where $X_0\in (a,b)\subseteq (0,1)$, $c>0$, and $B_t$ is the standard ...
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7 views

Is the result of a Monte-Carlo simulation of a continuous function and with continuous input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
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54 views

Modeling with Markov Chains and one-step analysis

I have set up the following model: Let $X_n$ be the number of heads in the $n$-th toss and $P(X_0=0)=1$. I can calculate the transition matrix $P$. Define $$ T=\min\{n\geq 0\mid X_n=5\}. $$ Then ...
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24 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
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How is the form of the transition functions of a process define as the solution of a stochastic differential equation?

I´m searching for a prove that solutions of SDE are markovian and i'm trying to find (or understand) in this case (SDE) what form have the transition functions associated to this kind of process. I´ve ...
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Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
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12 views

Stationary process vs stationary increments

Am I right that these are not the same, i.e. a stationary process need not have stationary increments and vice versa? example: Brownian motion is not a stationary process but it has stationary ...
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(joint) Functional CLT for partial sums and counting process

Assume you are given a sequence of random variables $(X_i)_{i\geq1}$. Assume moreover that they are sufficiently smooth, say $\mathbb E[X^2]<+\infty$. Define the diffusion-scaled partial sum as ...
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Joint convergence of stochastic processes

Suppose I have processes $X_n(t)\overset{d}{\longrightarrow} X(t)$, and $Y_n(t)\overset{p}{\longrightarrow} ct$ for some constant $c$. Then, can I conclude like in Slutsky's theorem that ...
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15 views

A clarification on $L_{loc}^2$ process and stochastic exponential

In the book by A. Pascucci (PDE and Martingale Methods in Option Pricing) I have found the following definition of $\mathbb{L}^2_{\text{loc}}$ process. Later (pp. 329-330) for a process ...
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Compound Poisson Process - calculate [on hold]

Compound Poisson Process $\{Y_t : 0 \le t \}$ with the intensity $\lambda$ there are breaks with distribution $exp(a)$, a>0. Calculate: $$P \{ Y_t >2 |N_t = 3\}$$. Any ideas? I know that $Y_t$ is ...
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46 views

Defining the states when we roll one single die repeatedly

We roll a single die and the game stops as soon as the sum of two successive rolls is either 5 or 7. We want to find the probability that the game stops at a sum of 5. It seems like Markov ...
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35 views

sigma algebra of a stopping time

Let $N$ be a stopping time. i.e $\{N=n\} \in \mathbb{f}_n \forall n$. $\mathbb{f}_n$ is the filtration. $\mathbb{f}_N=\{A\in \mathbb{f}, A\cap \{N=n\}\in \mathbb{f}_n \forall n\}$ is the sigma ...
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A functional of a Lévy process

Does anyone know if there are any papers/results on functionals of the type : $$\int_0^tp(X_s)ds$$ where $X$ is a Lévy process and $p$ is a polynomial. For example, is the distribution of such an ...
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19 views

Stock Price Dynamics correlated with Bond market returns

I am currently working on to derive the following form of the stock price dynamics: $$dS_t = S_t[(r_t + \psi\sigma_S)dt + \rho \sigma_S dz_{1t} + \sqrt{1-\rho^2}\sigma_S dz_{2t}$$ where the ...
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22 views

Adapted and progressive processes

Could you please help me proving rigorously the following fact from Mayer's book: (a) if $X_t$ is a process adapted with respect to filtration $\{\mathcal{F}_t\}_{t\ge 0}$ and for every ...
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31 views

Stochastic process on compact spaces

I just heard some strange reasoning that I would like to understand with your help, let me describe the situation (unfortunately, I hesitated to ask the lecturer about it, because I apparently lacked ...
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22 views

Existence of compensator process under the assumption of local integrability and finite variation

I am reading a proof regarding existence of compensators under the assumption of local integrability in which I don't quite understand: Definition: The compensator of a cadlag adapted process $X$ ...
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9 views

Different definitions of local p integrability for local martingales

When talking about cadlag (but not continuous) martingales and local martingales in the context of stochastic integration one can come across different definitions depending on the author. These are: ...
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$L^p$ integrable local martingale is still $L^p$ integrable when stopped at localizing stopping times.

Assume that $X$ is $L^p$ integrable for $1\leq p\leq \infty$ (i.e., for all $t$, $X_t\in L^p$) and is also a (Cadlag) local martingale. If $T_n$ is a localizing sequence of stopping times for $X$. Is ...
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40 views

What is “white noise” and how is it related to the Brownian motion?

In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering ...
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What is the benefit of stochastic models over deterministic models? [duplicate]

I have posted a similar question earlier and I guess this sounds naive to all of you, but nonetheless let me just ask: Consider I have a simple and deterministic model $M$, with a number of input ...
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1answer
12 views

Proving that $X^2- [X]$ is a local martingale given that $X$ is a cadlag locally square-integrable martingale

Suppose that $X$ is a cadlag locally square-integrable martingale. Let $[X]$ denote the quadratic variation of $X$. My textbook claims, by Ito's formula that $$ X^2 _t = X^2_0 + [X]_t + 2 ...
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53 views

Explain why $E(X)=1.65$ and $Var(X)=1.64$

Let $U$ be uniformly distributed on the interval [$\frac{1}{3},1$]. Let $X$ be a random variable such that the conditional distribution of $X$ given $U=p$ is Geometric with parameter $p$. (a) Find ...
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8 views

Stochastically removing edges from a complete graph; looking for info on this process

Given a complete graph of $n$ nodes and a time interval $T = \{t_1, \ldots, t_k\}$. Suppose each edge has a small probability $p$ of being removed from the graph at each time step and that this $p$ is ...
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22 views

Find the conditional and marginal densities?

I'm getting lost in notation here. We have $f_X(x)=xe^{-x}$ and we have a new random variable, $Y$, which is uniform from $(0,X)$. We want three things: Marginal density for $Y$ Conditional ...
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How to calculate the transition density for a multivariate jump process

I have the following stochastic process: $dX = (A-I)XdN$, where $X$ is a $2\times1$ vector of random variables, $A$ is a constant, real, symmetric, $2\times2$ matrix, $I$ is the identity matrix and ...
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48 views

Monte-Carlo simulation with sampling from uniform distribution

I used to work with Monte-Carlo simulations for a while. In my case, I generated random data for a variety of input parameters according to uniform distributions (with non-negative support), say for ...
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Some problems about symmetric convolution semigroup on the unit circle

These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes". Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
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1answer
15 views

Probability Brownian motion is positive at two points

Let $0<s<t$ and $(B_r)_r$ is Brownian motion. Does anybody know what $P(B_s>0,B_t>0)$ is? I think I remember it was some $arctan$-law but I don't know the exact form. So I do not need a ...
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How can we evaluate the material derivative of the velocity of an particle by means of an Itō formula?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(B_t)_{t\ge 0}$ be a $\mathbb R^d$-valued Brownian motion with respect to ...
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local martingales/ Ito formula

I have a problem with following task. Find $(A_t)_{t\ge0}$ a process of bounded variation on bounded intervals, such that $A_0=0$ and process $M_t=W_tsin(\int^t_0W_s^3dW_s)-A_t$ is a local martingale. ...
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24 views

Let $X(t) = e^{r(T-t)}/S(t)$. Find the SDE of $X(t)$ provided that $S(t)$ satisfies the BSM model.

This is the last part to a 3 part question! I am nearly done going through the questions I had difficulty with while studying, again, anyone's help would be greatly appreciated! Let $X(t) = ...
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20 views

If $S(t)$ is the stock price that satisfies BSM model in SDE form how can I derive an SDE for $S^n (t)$ for some positive integer n

If $S(t)$ is the stock price that satisfies BSM model in SDE form where $dS(t) = \mu S(t) dt + \sigma S(t) d W(t)$ where $\mu >0$ and $\sigma>0$ are two constants. how can I derive an SDE for ...
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22 views

An example of a reversible but reducible Markov chain

The reversibility of a Markov chain is defined in the following way with some basic propositions. Unfortunately all examples of reversible Markov chains shown in my textbook so far are irreducible, ...
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18 views

Product sigma algebra in the countable case

Let $(\Omega_i,\mathcal{F_i})_{i \in \mathbb{N}}$ be the countable family of measurable spaces.Then I was wondering whether the canonical product sigma algebra on the product space that is for example ...
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25 views

derive integration by parts for a stochastic integral

The question is to show the following identity: $\int_{0}^{T}tdW(t) = TW(T)-\int_{0}^{T}W(t)dt$ This can be done quite easily with ito's however the question explicitly says to show the identity ...
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1answer
36 views

Average of Monte Carlo simulations of continuous functions again continuous?

I hope the following question is clear: Suppose, we have a continuous functions $f:\mathbb{N}^2 \rightarrow \mathbb{N}$. Now, suppose we run Monte Carlo simulations on the function, where the input ...
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1answer
35 views

calculate mean and variance from multivariate probability-generating function in random walks

Suppose in a biased random walk, $r(i,n)$ is the probability that a particle appears at position $i$ at time $n$. The corresponding probability generating function is $$ R(z,s)=\sum_{n=0}^\infty ...
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How can we prove that the derivative of a generalized Hilbert space valued Brownian motion is a Gaussian white noise?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\lambda$ be the Lebesgue measure on $[0,\infty)$ $\mathcal D:=C_c^\infty([0,\infty))$ and $\mathcal D'$ be the dual space of ...
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How can I solve $\phi_1$ and $\phi_2$?

Let be $$\rho(1)=\frac{\phi_1}{1-\phi_2}, \rho(2)=\frac{\phi_1^2+\phi_2(1-\phi_1)}{1-\phi_2}$$ How can I solve $\phi_1,\phi_2$? My idea: $\rho(1)=\frac{\phi_1}{1-\phi_2}\Leftrightarrow ...
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Locally square integrable (local) martingales

I'm reading Protter and sometimes he says "locally square integrable martingale", and sometimes he says "locally square integrable local martingale", and I wonder if these two are the same. Protter's ...
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Formula for contingent claim similar to European call option but with two dates for option to buy

So in a normal European call option with one maturity date, you'd buy a share of a stock if the price of the stock at the maturity date was higher than the exercise price. How would you come up with a ...
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1answer
45 views

Random walk on $\mathbb Z/m\mathbb Z$ converges to uniform distribution

Let $(X_t)_t$ be the standard continuous time random walk on $\mathbb Z/m\mathbb Z$ with $X(0)=[0]$ almost surely, then I want to show that for $t \rightarrow \infty$, $\lim_{t \rightarrow \infty} ...
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29 views

heuristic for expected number of visits random walk

What is the heuristic argument that explains why, on $\mathbb{Z}^d$, $d \geq 3$, the expected number of visits of a random walk starting from the origin at $x$ is of order $$ O(|x|^{2-d})? $$
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48 views

Prove that the limit in probability of normally distributed random variables is normally distributed, too [duplicate]

Let $X_n\sim\mathcal N_{\mu_n,\sigma_n^2}$ for some $(\mu_n,\sigma_n^2)\in\mathbb R\times(0,\infty)$ and $X$ be a real-valued random variable with $$X_n\stackrel{\text{in probability}}\to X\;.\tag 1$$ ...
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Convex closure of the support of a Levy Process

Given a Levy Process $X_{t}$ on a filtered Probability Space $(\Omega,\mathcal{F},\mathcal{F}_{t},P)$ with distribution-function $F_{t}$ for $X_{t}$. We look now for the cumulant transform $\phi_{1}$ ...
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28 views

How can we prove that the generalized stochastic process induced by a real-valued Brownian motion is Gaussian?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...